Solve for $x$ and $y$ using elimination. $\begin{align*}-7x+4y &= -8 \\ 5x-6y &= 1\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $2$ $\begin{align*}-21x+12y &= -24\\ 10x-12y &= 2\end{align*}$ Add the top and bottom equations. $-11x = -22$ Divide both sides by $-11$ and reduce as necessary. $x = 2$ Substitute $2$ for $x$ in the top equation. $-7( 2)+4y = -8$ $-14+4y = -8$ $4y = 6$ $y = \dfrac{3}{2}$ The solution is $\enspace x = 2, \enspace y = \dfrac{3}{2}$.